These data are required to satisfy a long list of coherence diagrams see 55 Ap

These data are required to satisfy a long list of

This preview shows page 158 - 163 out of 183 pages.

These data are required to satisfy a long list of coherence diagrams, see [ 55 , Ap- pendix C] for details. Definition A.24. A symmetric monoidal 2-functor between two symmetric monoidal bicategories B and B 0 consists of a 2-functor H : B -→ B 0 of the underlying bicat- egories together with the following data: (a) Equivalence natural 2-transformations 3 χ : 0 ( H ( · ) × H ( · ) ) = ⇒ H ◦ ⊗ and ι : 1 0 = ⇒ H (1) , where here we consider 1 as a 2-functor from the bicategory with one object, one 1-morphism and one 2-morphism to B . (b) The three invertible modifications H ( a ) 0 ( H ( b ) 0 H ( c ) ) H ( a ) 0 H ( b c ) ( H ( a ) 0 H ( b ) ) 0 H ( c ) H ( a ( b c ) ) H ( a b ) 0 H ( c ) H ( ( a b ) c ) id 0 χ χ α 0 χ 0 id χ H ( α ) Ω (A.25) 3 We fix again adjoint inverses and the adjunction data. 152
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Chapter A: Symmetric monoidal bicategories H (1) 0 H ( a ) H (1 a ) 1 0 0 H ( a ) H ( a ) χ H ( λ ) ι 0 id λ 0 Γ and H ( a ) 0 1 0 H ( a ) 0 H (1) H ( a ) H ( a 1) id 0 ι χ ρ 0 H ( ρ ) Δ (c) An invertible modification H ( b a ) H ( b ) 0 H ( a ) H ( a b ) H ( a ) 0 H ( b ) H ( β ) Υ χ β 0 χ (A.26) These data are required to satisfy a long list of coherence conditions, see [ 55 ] and references therein for details. Our definition of symmetric monoidal transformations differs slightly from [ 55 ]. The definition we give is tailored to the application in functorial field theories. In contrast to the definition given in [ 55 ], we require the appearing modifications to be invertible. However, the 2-morphisms corresponding to the underlying natural transformations are not invertible in our definition, so our definition is also weaker than the definition given in [ 55 ]. Definition A.27. A natural symmetric monoidal 2-transformation between sym- metric monoidal 2-functors H , K : B -→ B 0 consists of a natural 2-transformation θ : H = ⇒ K of the underlying 2-functors together with invertible modifications H ( a b ) H ( a ) 0 H ( b ) K ( a b ) K ( a ) 0 H ( b ) K ( a ) 0 K ( b ) θ χ H θ 0 id id 0 θ Π χ K (A.28) and 1 0 K (1) H (1) ι K ι H M θ (A.29) 153
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Chapter A: Symmetric monoidal bicategories which satisfy the following coherence conditions expressed as equalities between 2- morphisms (omitting tensor product symbols on objects and 1-morphisms to stream- line the notation): K ( a ) ( K ( b ) H ( c ) ) K ( a ) ( K ( b ) K ( c ) ) ( ( K ( a ) K ( b ) ) K ( c ) ( K ( a ) K ( b ) ) H ( c ) K ( ab ) K ( c ) K ( ab ) H ( c ) ( K ( a ) H ( b ) ) H ( c ) K ( ( ab ) c ) H ( ab ) H ( c ) ( H ( a ) H ( b ) ) H ( c ) H ( ( ab ) c ) K ( a ( bc ) ) H ( a ) ( H ( b ) H ( c ) ) H ( c ) H ( bc ) H ( a ( bc )) θ α 0 χ K α 0 θ χ K χ K θ Π θ Π 0 id K ( α ) θ χ H Ω H θ α θ χ H α 0 H ( α ) θ χ H χ H θ k K ( a ) ( K ( b ) H ( c ) ) K ( a ) ( K ( b ) K ( c ) ) ( K ( a ) K ( b ) ) K ( c ) ( K ( a ) K ( b ) ) H ( c ) K ( a ) ( K ( b ) K ( c ) ) K ( ab ) K ( c ) ( K ( a ) H ( b ) ) H ( c ) K ( a ) K ( bc ) K ( ( ab ) c ) K ( a ) ( H ( b ) H ( c ) ) ( H ( a ) H ( b ) ) H ( c ) K ( a ) H ( bc ) K ( a ( bc ) ) H ( a ) ( H ( b ) H ( c ) ) H ( a ) H ( bc ) H ( a ( bc ) ) θ id 0 Π α 0 χ K χ K α 0 α 0 χ K χ K Ω K α 0 ? θ α 0 χ K Π K ( α ) θ Φ 0 χ H θ α 0 α 0 ? θ χ H θ θ χ H θ (A.30) 154
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Chapter A: Symmetric monoidal bicategories K (1) H ( a ) K (1) K ( a ) H (1) H ( a ) H (1 a ) K (1 a ) 1 0 H ( a ) H ( a ) K ( a ) 1 0 K ( a ) θ Π χ K θ χ H θ H ( λ ) K ( λ ) ι H λ 0 θ θ λ 0 θa λ 0 Γ H θ k K (1) H ( a ) K (1) K ( a ) H (1) H ( a ) K (1 a ) 1 0 H ( a ) K ( a ) 1 0 K ( a ) θ Φ 0 χ K θ M - 1 0 id K ( λ ) ι H ι K θ Γ K λ 0 ι K (A.31) K ( a )1 0 K ( a ) H (1) K ( a ) H ( a )1 0 H ( a ) H (1) K ( a ) K (1) H ( a ) H ( a 1) K ( a 1) K ( a ) ι H θ ρ 0 ρ 0 θ θ ι H Π χ H θ χ K θ θ ρ 0 H ( ρ ) Δ - 1 H θ ρ θ K ( ρ ) ι θ k 155
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Chapter A: Symmetric monoidal bicategories K ( a )1 0 K ( a ) H (1) K ( a ) K ( a ) K (1) H ( a ) K ( a 1) K ( a ) ι K ι H θ ρ 0 K ( ρ ) id id 0 M - 1 χ K θ θ K ( ρ ) Δ - 1 K (A.32) and H ( b ) H ( a ) H ( ba ) H ( b ) H ( a ) H ( ab ) K ( ba ) K ( a ) K ( b ) K ( ab ) χ H Υ
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